[HN Gopher] Problems in Linear Algebra (1978), a book of solved ...
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Problems in Linear Algebra (1978), a book of solved problems
Author : happy-go-lucky
Score : 141 points
Date : 2022-06-02 12:02 UTC (10 hours ago)
(HTM) web link (archive.org)
(TXT) w3m dump (archive.org)
| isaacfrond wrote:
| Boy, you've got a lot of problems.
| iqkznnft wrote:
| 99, to be precise.
| stiff wrote:
| Paul Halmos wrote a book I like a lot titled "Linear Algebra
| Problem Book" that teaches you linear algebra through a sequence
| of problems - its different from the one posted here in that the
| problems are often preceded by 2-3 pages of discussion/motivation
| and almost all of them are conceptual and interesting.
| ogogmad wrote:
| Exercise: Let M be a square complex matrix satisfying M = M^T,
| and furthermore assume that M is not singular. Prove that there
| exists a matrix K such that K^2 = M and K = K^T.
|
| Non-solution: You are essentially being asked to show that given
| an invertible complex-symmetric matrix, you can find a complex-
| symmetric square root. You might be tempted to use the Taylor
| series of sqrt(z) centred at some z_0 to obtain K, but this won't
| work if M's eigenvalues can't be fit inside a half-plane not
| include the complex number 0. An example of this is when M is
| [?]1 0 [?] [?] [?] [?]0 -1[?]
|
| Another non-solution: You might attempt to extend the spectral
| theorem to complex-symmetric matrices, but the only such
| extension I know of is complicated, and results in a block-
| diagonal (but not necessarily diagonal) canonical form.
|
| Solution: To actually extend the sqrt function from the complex
| numbers to matrices, use the Jordan canonical form of M. Then you
| only need to take the sqrt's of each Jordan block. Note that this
| sqrt won't exist if a Jordan block is non-zero and nilpotent, so
| we require M to be non-singular. This obtains K. Finally, observe
| that there exists a polynomial p such that p(M) = K (which can be
| obtained via Hermite interpolation), and therefore K^T = p(M)^T =
| p(M^T) = p(M) = K.
| WoahNoun wrote:
| There's a geometric solution as well!
|
| Since M is invertible and symmetric. It acts on R^N by
| independently scaling the coordinates of an orthogonal basis.
| Eg, M = O^T*A*O where O is orthogonal and A is a diagonal
| matrix. Let sqrt(A) be the diagonal matrix with entries equal
| to the square root of the corresponding entry in A. Then K =
| O^T*sqrt(A)*O is symmetric and K^2 =
| (O^T*sqrt(A)*O)*(O^T*sqrt(A)*O). Since O is orthogonal O^T*O =
| I, and K^2 = O^T*A*O = M.
| ogogmad wrote:
| That doesn't work. It's complex symmetric, but not real
| symmetric or Hermitian, so you can't use the spectral
| theorem.
|
| This statement here: It acts on R^N by
| independently scaling the coordinates of an orthogonal basis
|
| is precisely the spectral theorem, which you can't use here.
|
| [edit]
|
| Also, the corresponding claim for Hermitian (and real
| symmetric) matrices isn't true either. If an eigenvalue of a
| Hermitian matrix H is negative, then no square root of it can
| be Hermitian (or real symmetric).
| WoahNoun wrote:
| Ah I missed the complex part. Clearly a complex matrix
| doesn't act on R^N by scaling the coordinates.
| vippy wrote:
| I've got 1938 problems but proving that the subspace L of the
| unitary space R_n has an orthogonal basis consisting of the
| eigenvectors of the transformation ph ain't one.
| WoahNoun wrote:
| Skimming through there are some interesting problems (the
| fibonacci sequence generated by determinants of nxn matrices with
| -1/1 on the lower subdiagonal/upper subdiagonal is pretty cool).
| But a lot of these problems just look like super tedious
| calculations.
| jedimastert wrote:
| > a lot of these problems just look like super tedious
| calculations
|
| Analog Stack Overflow?
| xdavidliu wrote:
| stackoverflow is a site for programming questions. Where does
| one see tedious mathematical calculations on there?
| burnished wrote:
| The math related stackoverflows
| WoahNoun wrote:
| Mathoverflow is run by StackExchange.
|
| https://mathoverflow.net/
| cpp_frog wrote:
| That's the general style of the book, but further along the way
| it has problems that require proving statements by referring to
| the properties or definitions of the actual structures involved
| (vector spaces).
| begriffs wrote:
| I put together a list of the best linear algebra books,
| classified by type. Proskuryakov's practice book is one of them.
| Here's the full list:
|
| https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo...
| cpp_frog wrote:
| I have the version of this book in Spanish (Editorial MIR), the
| translated title is _2000 Problems in Linear Algebra_. This
| publishing house (MIR) doesn 't exist anymore but now goes under
| the name of URSS [0] and publishes modern versions of these same
| books, at least in Spanish and Russian. It has several books on
| problem solving, which was probably a pathway to demonstrating
| true "elite" status in eastern europe at the time (hence the
| known romanian, hungarian and russian way of teaching). It also
| contains several books on theory too, be it mathematics, physics
| or engineering. The important thing with textbooks on theory is,
| unlike the custom of modern textbooks to have few problems and
| often disconnected or that require to make a few "leaps" in the
| sense that the exposition does not clearly pave a way to solve
| harder problems, the amount of exercises is large and they
| frequently build on top of one another, thus making learning much
| more enjoyable in my opinion.
|
| Some books also worth looking at in mathematics:
|
| (1) _Anti-Demidovich series_
|
| (2) _Combinatorial Analysis_ , Ribnikov
|
| (3) Problem series by Suprun
|
| (4) _Probability Theory_ , Zolotarievskaya
|
| (5) _Discrete Mathematics Problems_ , Evnin
|
| (6) _Theory of Surfaces_ , Finikov
|
| ... and several other problem book series for university
| mathematics.
|
| Engineering (mechanics):
|
| (1) _Anti-Mescherski series_
|
| (2) _Theoretical Mechanics_ , V. M. Starzhinski
|
| (3) _Strength of Materials_ , Stiopin
|
| (4) _Problems in Strength of Materials_ , Volmir
|
| [0] https://urss.ru/cgi-
| bin/db.pl?lang=sp&blang=en&page=Bookstor... (apparently the
| headquarters are in Peru now)
| lfmunoz4 wrote:
| mananaysiempre wrote:
| Note though that URSS is a ruthlessly commercial enterprise,
| and in the absence of any real competition, their printing is
| frequently bad, covers tasteless, translation and editing
| passable (which is better than can be said for more generalist
| Russian publishing houses' attempts at technical literature)
| but only just, and prices high. (Before this February, the EUR
| prices before shipping were about 250% of the already-
| substantial RUB ones per the official conversion rates;
| nowadays I don't know.) If you can at all help it, try the
| MCCME / IUM press and bookshop[1] first; unfortunately they do
| not deal in second-hand books (or rather they do, a bit, but
| people will pick those up in person in a day or two, before
| anybody even bothers entering them into the online catalogue).
|
| Let me also add a recommendation for the calculus problems book
| by Gunther (Gyunter) and Kuzmin (I don't think it was ever
| translated, but how much translation do you really need in
| one?). It will not _train_ you--every idea occurs once or maybe
| twice; it is not a book of exercises. (The joke goes that
| Demidovich is G &K with every problem repeated ten times.) But
| it is a book of _problems_ , and it will teach you.
|
| [1] https://biblio.mccme.ru/
| azalemeth wrote:
| At a glance, probably Western equivalent of these are Schaum's
| Outline Series, which start by telling you something, giving
| you some worked examples on it, and then setting you lose on
| them. It's a great way to learn, quickly, particularly for a
| student.
|
| I remember distinctly learning (as a teenager) Schaum's Outline
| Series of Vector Calculus teaching me quickly and in detail
| "grad div curl and all that" very effectively, so much so that
| I got basically 100% on all of my A-level maths and further
| maths papers because, well, if you _can_ do a big book of
| problems like that, then exams really are just more of the
| same. At the time, I absolutely loved them -- big brown books,
| some of which are currently by my ankles.
|
| I'm not sure it actually _helped_ that much _beyond_ early
| university though. Exams select for a very specific skill. What
| isn 't taught is _why_ something was discovered, or is useful,
| or _how_ an idea came to be. Many more advanced ideas in
| mathematics are downright bizarre and it 's the basic _idea_
| that you need to be able to come up with something similar to,
| not necessarily the detail.
| rg111 wrote:
| By that book, do you mean the one authored by Spiegel?
|
| I went over that book fully. From cover to cover. It was
| absolutely fantastic, and as good as a "problems book" can
| be.
|
| I loved it and would recommend it to anyone who is looking to
| get their feet wet with problems in Vector Calculus.
|
| Although it is neither a rigorous treatment (look into
| Arfken, Weber, Harris for that), nor it is greatly _healpful_
| pedagogically (look into Mary L. Boas for that), nor is it
| easy (look into Riley, Hobson, Bence for that).
|
| The greatest intro material into Linear Algebra still remains
| the one by Gilbert Strang.
| cpp_frog wrote:
| I completely agree with you. A corollary of my comment is
| that while not necessarily providing theoretical
| justification, sometimes doing this whole problem solving
| thing helps seeing why a field of knowledge came to study
| some of its problems, fast. For example, I have a strong
| abstract foundation in Functional Analysis, Measure Theory
| and Optimization and my field is Numerical Analysis of PDE. I
| didn't understand 1. some mechanical properties of materials,
| 2. some properties of tensors and differential geometry that
| I needed. But I didn't have so much time to devote and work
| out the proofs for all the abstract apparatus on my own. So I
| took an engineering textbook on tensors, some notes on
| relativity (because they use tensors and Christoffel symbols)
| and in a few weeks I had almost understood everything I
| needed.
|
| On the other hand, there is an extremely rigorous book with
| all the proofs for differential geometry, by Kennington. It
| has 2400 pages. For tensors I would guess that I need more
| Algebra, specifically, modules. And to review smoothness and
| differentiability classes.
| dls2016 wrote:
| Re: Kennington... I check in every year or so to see his
| progress.
|
| For those interested: topology.org
| pvg wrote:
| For easier googling, the names of some of these authors in
| their slightly different English transliteration:
|
| Rybnikov
|
| Zolotarevskaya
|
| Stepin
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